On the Non-existence of Equilibrium: From Thornton to Arrow

Abstract

Among the many contributions to economic science made by Arrow, one of the most important is certainly the proof of the existence of an equilibrium for a competitive economy.¹ The case where no equilibrium exists even though indifference curves, production functions, and so on, are fairly well behaved is a useful one to show the necessity of proving the existence of equilibrium. Mill (1869) indicates that one of the first examples of the non-existence of equilibrium consists of the counter examples to equilibrium theory given in W. T. Thornton’s On Labour (1869) though Thornton himself was concerned not so much with the non-existence of equilibrium as the possibility of trade at disequilibrium prices. The Thornton-Mill examples of the non-existence of equilibrium are remarkable because they are due to the discontinuity of demand curves; other unsuccessful attempts to show the non-existence of equilibrium failed because of their assumption of continuity. The Thornton-Mill examples, as well as the example of Wald (1951), are, however, not so serious to equilibrium theory, if we consider, not the Walrasian tâtonnement with recontract, but the non-tâtonnement without recontract. From such non-tâtonnement point of view, a truly important example of the non-existence of equilibrium is the one given by Arrow, that is, the case where a Pareto optimal allocation cannot be viewed as a competitive equilibrium.²

I must thank Professor K. J. Arrow for invaluable suggestions and warm encouragement given to my early studies of general equilibrium theory. It was in 1957 when I was a first year graduate student at Tokyo that I dared to write to him to discuss an alternative proof of the existence of equilibrium based on Pareto optimality of a competitive equilibrium, which was later published in Metroeconomica (1960). Subsequently, I joined the Office of Naval Research project at Stanford, where I began my studies of the non-tâtonnement stability problem.